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Angle-Angle-Side (AAS) Postulate
Before Section 3.3 One more way to prove triangles congruent: Angle-Angle-Side (AAS) Postulate If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of a second triangle, then the two triangles are congruent. A B C X Y Z
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Definition of midpoint
C B D M F Statements Reasons 1. 2. 3. 4. 5. 6. B C Given D F Given Given DM = MF Definition of midpoint Definition of congruent segments ∆BDM ∆CFM AAS
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3.3 CPCTC and Circles Objectives: Use CPCTC in proofs
Know and use basic properties of circles
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Congruent triangles: triangles whose corresponding parts are congruent.
CPCTC Corresponding Part of Congruent Triangles are Congruent. This can be used in a proof only AFTER triangles have been proven congruent.
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Notation: circle C or ⨀C.
Circle: the set of points in a plane equidistant from the one point, the center. Notation: circle C or ⨀C. Radius: the distance from the center of the circle to any point on the circle. Theorem 19: All radii of a circle are congruent. C
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Circumference of a Circle: C = 2r
Area of a Circle: A = r2 Example 1: Find the circumference and area of a circle with a radius of 12 units.
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N M O L Example 2: Given: ⨀O 𝑁𝑂 ⊥ 𝐿𝑀 Prove: ∆NOL ∆NOM
𝑁𝑂 ⊥ 𝐿𝑀 Prove: ∆NOL ∆NOM L M O N Statements Reasons 1. ⨀O 1. Given 2. 3. 4. 5. All radii of a circle are congruent Given NOL and NOM are right angles Definition of perpendicular NOL NOM All right angles are congruent ∆NOL ∆NOM SAS
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Given: Z is the midpoint of Y and W are complementary to V Prove: W
Example 3: Given: Z is the midpoint of Y and W are complementary to V Prove: W Z V X Y Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. 7. 8. Z is the midpoint of WZ = ZY Definition of midpoint Definition of congruent Y and W are comp. to V Given Y W Congruent Complements Theorem VZY WZX Vertical Angles Theorem ∆VZY ∆XZW ASA CPCTC
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N K P M L Example 4: Given: ⨀P Prove:
Statements Reasons 1. ⨀𝑃 1. Given 2. 3. 4. 5. All radii of a circle are congruent KPL NPM Vertical Angles Theorem ∆VZY ∆XZW SAS CPCTC
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T Q R S Example 5: Given: ⨀Q RT = TS Prove: TRQ TSQ
Statements Reasons 1. ⨀Q 1. Given 2. 3. 4. 5. 6. All radii of a circle are congruent RT = RS Given Definition of congruent ∆TRQ ∆TSQ SSS TRQ TSQ CPCTC
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F A E B C D Example 6: Given: C is the midpoint of AC = CE
Prove: ∆ABF ∆EDF A E B C D Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. 7. 8. FCA and FCE are right angles Definition of perpendicular FCA FCE All right angles are congruent Reflexive Property AC = CE Given Definition of congruent ∆FCA ∆FCE SAS A E CPCTC Continued on next slide
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F A E B C D Example 6: Given: C is the midpoint of AC = CE
Prove: ∆ABF ∆EDF A E B C D Statements Reasons 9. 10. 11. 12. 13. 14. C is the midpoint of Given BC = CD Definition of midpoint Definition of congruent Subtraction Property Given ∆ABF ∆EDF SAS
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A B C G H D E F Example 7: Definition of bisect Definition of bisect
Statements Reasons 1. 1. Given 2. 3. 4. 5. 6. Definition of bisect Definition of bisect AGD BGE Vertical Angles Theorem ∆AGD ∆BGE SAS CPCTC Continued on next slide
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A B C G H D E F Example 7: Definition of bisect Definition of bisect
Statements Reasons 7. 7. Given 8. 9. 10. 11. 12. 13. Definition of bisect Definition of bisect BHE CHF Vertical Angles Theorem ∆BHE ∆CHF SAS CPCTC Transitive Property
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